Any light beam can be described by a four-element Stokes vector, which describes the polarization of the light beam and its intensity. For an optical measurement that conserves incident photon energy, any interaction of a sample with a polarized light beam described by a Stokes vector will result in another light beam, also described by a four-element Stokes vector. In the general case, sixteen parameters (a 4.times.4 Mueller matrix) are required to describe this interaction.
In most cases, these sixteen elements are not unique. For example, if the sample is isotropic and the experiment is set up as a reflection ellipsometer, then only three Mueller matrix elements are unique. However, if the sample is not isotropic, or if there are optical elements other than an isotropic sample between the polarization state generator (PSG) and the polarization state detector (PSD), then a larger number of the Mueller matrix elements are non-zero and are unique.
Reflection ellipsometers are optical instruments commonly used to characterize thin films and the optical properties of bulk materials. A defining characteristic of these instruments is that a light beam from the PSG is reflected from the sample surface at a large angle of incidence. The light beam can be either of a single wavelength (usually using a laser as an illumination source), or spectroscopic (usually using a white light source in conjunction with a monochromator). The PSG and the PSD typically contain linear polarizers and possibly optical compensating elements such as quarter-wave plates, Fresnel rhombs, or photoelastic modulators. Most ellipsometers measure one to three independent parameters that are related to linear combinations of Mueller matrix elements. For isotropic samples where there is no significant perturbation of the light beam between the sample and either the PSG or the PSD (such as vacuum chamber windows), two or three independent parameters are often sufficient to characterize the sample. However, if the sample is anisotropic, or if there are additional optical elements between the sample and the PSG and the PSD, then these instruments are not sufficient to characterize the sample. For example, the rotating analyzer ellipsometer (which has a stationary polarizer as the PSG and a rotating polarizer as the PSD), and the rotating polarizer ellipsometer (which has a stationary polarizer as the PSD and a rotating polarizer as the PSG) each measure two independent parameters.
Polarization modulation ellipsometers (which contain a polarizer-photoelastic modulator pair in either the PSG or the PSD, and a polarizer in the other), also measure only two independent parameters, but the third parameter can be measured using a different azimuthal orientation of the polarizer (S. N. Jasperson and S. E. Schnatterly, "An Improved Method for High Reflectivity Ellipsometry Based on a New Polarization Modulation Technique," Rev. Sci. Instrum. 40, 761-767 (1969); 41 152 (1970).); (B. Drevillon, J. Perrin, R. Marbot, A. Violet, and J. L. Dalby, "Fast Polarization Modulated Ellipsometer Using a Microprocessor System for Digital Fourier Analysis," Rev. Sci. Instrum. 53, 969-977 (1982).). The two-channel spectroscopic polarization modulation ellipsometer (which contains a polarizer-photoelastic modulator pair as the PSG and a Wollaston prism as the PSD), (G. E. Jellison, Jr. and F. A. Modine, "Two-Channel Polarization Modulation Ellipsometer," Appl. Opt. 29, 959-973 (1990).) measures three independent parameters simultaneously when the azimuthal angle of the PSG is set to .+-.22.5.degree. or to .+-.67.5.degree. with respect to the plane of incidence defined by the sample surface.
Transmission ellipsometers are similar to reflection ellipsometers, except that the incident light passes through the sample. This complicates the analysis since sample thickness variations can partially depolarize the incident beam. The subclass of transmission ellipsometers encompasses several instruments that are designed for certain specific measurement tasks. Dichrographs, for example, are used to measure the difference in optical absorption between the fast and slow axis directions of linearly dichroic materials or to measure the difference in optical absorption for left-and right-circularly polarized light by chiral optical materials.
Other instruments measure linear or circular birefringence. These instruments typically measure only one element of the sample Mueller matrix, which is then interpreted to obtain a specific piece of information. Sacharimeters, for example, measure the rotation of the plane of polarization of a linearly polarized beam by a solution, and they are employed to measure sugar concentration. A sacharimeter can be as simple as a light beam from a laser passing through a polarizer, the sample, another polarizer and then detected, where the azimuthal angle of one of the polarizers is changed to minimize the light reaching the detector. The sugar concentration is determined by measuring the extinction position with the sample in place compared to the extinction position when the sample is removed.
Perturbation spectrometers of various types are also optical instruments that can be generally classified as ellipsometers. In such spectrometers, an optical anisotropy is induced in a sample by applying a perturbation such as an electrical or magnetic field or mechanical stress. The methods and instrumentation of ellipsometry are then used to measure the perturbed optical properties. Instruments that measure Faraday rotation are one such example.
Instruments that can measure the entire Mueller matrix of a sample have been proposed and built previously. One such instrument is the two-compensator ellipsometer (2CE) (P. S. Hauge, "Recent Developments in Instrumentation in Ellipsometry," Surf Sci. 96 108-140 (1980).). It uses rotating fixed-wavelength compensators, so significant data corrections are required to operate the instrument at wavelengths other than the design wavelength. The term "generalized ellipsometer" has been used to describe these and other instruments that measure more parameters than are measured with conventional ellipsometers.
Ellipsometers based on two polarizer-photoelastic modulator pairs have been postulated by Hauge (P. S. Hauge, "Recent Developments in Instrumentation in Ellipsometry," Surf Sci. 96 108-140 (1980).) and by Anderson (Richard Anderson, "Measurement of Mueller Matrices," Appl. Opt. 31, 11-13 (1992).), which in principal would be capable of measuring eight normalized Mueller matrix elements. However, prior to our invention, herein described, no one has described an instrument based on two free running PEMs due to the instrumental complexity and the difficulty of interpreting the waveform. If such an ellipsometer could be built, all 15 normalized Mueller matrix elements could be measured with four measurements at different azimuthal angles of the PSG and the PSD.
Thus, prior to our invention, there has been no practical realization of an ellipsometer based upon two free-running PEMs. The PEMs are resonant devices (with a Q often exceeding 10,000), so their operating frequencies and relative phases are not settable but rather determined from other factors such as the shapes of the PEMs and their operating temperatures. Hereinbelow, we describe an apparatus and method for analyzing the waveform from two free-running photoelastic modulators, thereby providing a realizable and accurate two-modulator generalized ellipsometer.